Break a loop animation (Make Cyclic) to start a new one
I guess you will have to do it by the NLA Editor: push your first animation in the NLA Editor (down arrow), open the N menu of the NLA Editor > Animation Tab > Action Clip > Playback Settings > Repeat as much as you want, come back to the Dopesheet in Action Editor mode, create your second animation, then push it in the NLA Editor, by default it will appear on a second track above the first one, then move the clip of this second animation where you want it to interrupt your loop.